Munich Personal RePEc Archive

Global Solutions to DSGE Models as a

Perturbation of a Deterministic Path

Ajevskis, Viktors

Bank of Latvia, Riga Technical University

8 April 2014

Online at https://mpra.ub.uni-muenchen.de/55145/

MPRA Paper No. 55145, posted 04 Apr 2016 17:08 UTC

Global Solutions to DSGE Models as a

Perturbation of a Deterministic Path

Viktors Ajevskis∗

Bank of Latvia and Riga Technical UniversityViktors.Ajevskis@bank.lv

April 8, 2014

Abstract

This study presents an approach based on a perturbation techniqueto construct global solutions to dynamic stochastic general equilibriummodels (DSGE). The main idea is to expand a solution in a series of pow-ers of a small parameter scaling the uncertainty in the economy arounda solution to the deterministic model, i.e. the model where the volatilityof the shocks vanishes. If a deterministic path is global in state variables,then so are the constructed solutions to the stochastic model, whereasthese solutions are local in the scaling parameter. Under the assump-tion that a deterministic path is already known the higher order terms inthe expansion are obtained recursively by solving linear rational expecta-tions models with time-varying parameters. The present work proposes amethod rested on backward recursion for solving this type of models.

Key words: DSGE, perturbation method, rational expectations models withtime-varying parameters, asset pricing model

1 INTRODUCTION

Perturbation methods are the most widely-used approach to solve nonlinearDSGE models owing to their ability to deal with medium and large-size modelsfor reasonable computational time. Perturbations applied in macroeconomicsare used to expand the exact solution around a deterministic steady state inpowers of state variables and a parameter scaling the uncertainty in the economy.The solutions based on the Taylor series expansion are intrinsically local, i.e.they are accurate in some neighborhood (presumably small) of the deterministicsteady state. Out of the neighborhood, for example, in the case of sufficiently

∗I thank the WGEM ECB participants, Rudolfs Bems and Anton Nakov for useful com-ments and suggestions. The opinions expressed in this article are the sole responsibility of theauthor and do not necessarily reflect the position of the Bank of Latvia.

1

large shocks (or under the initial conditions that are far away from the steadystate) the approximated solution can imply explosive dynamics, even if theoriginal system is still stable for the same shocks (or initial conditions) (Kim,Kim, Schaumburg, and Sims (2008); Den Haan, and De Wind (2012)).

This study presents an approach based on a perturbation technique to con-struct global solutions to DSGE models. The proposed solutions are representedas a series in powers of a small parameter σ scaling the covariance matrix ofthe shocks. The zero order approximation corresponds to the solution to thedeterministic model, because all shocks vanish as σ = 0. Global solutions todeterministic models can be obtained reasonably fast by effective numericalmethods1 even for large size models (Hollinger (2008)). For this reason the nextstages of the method are carried out under the assumption that the solution tothe deterministic model under given initial conditions is known.

The higher-order systems depend only on quantities of lower orders, thereforecan be solved recursively. The homogeneous part of these systems is the samefor all orders and depends on the deterministic solution. Consequently, eachsystem can be represented as a rational expectation model with time-varyingparameters. In the case of rational expectations models with constant parame-ters the stable block of equations can be isolated and solved forward. This is notpossible for models with time-varying parameters. The present work proposesa method for solving this type of models. The method starts with finding afinite-horizon solution by using backward recursion. Next we prove that undercertain conditions as the horizon tends to infinity the finite-horizon solutionsapproach to a limit solution that is bounded for all positive time.

If the parameter σ is small enough, then the solutions obtained are close tothe deterministic solution. At the same time, whenever the deterministic solu-tion is global in state variables so is the approximate solution to the stochasticproblem. For this reason, we shall call this approach semi-global, whereas theperturbation methods based on series expansion around the steady state will bereferred to as local. In contrast to the solutions obtained by the local perturba-tion methods, the solutions provided by the semi-global method inherit “global”properties, such as monotonicity and convexity, from the exact solution and thuscannot explode by construction.

We apply the method to the asset pricing model of Burnside (1998). Sincethe model has a closed-form solution we can check the accuracy of an approx-imate solution against the exact one. We compare the accuracy of the secondorder solution of the semi-global method with the local Taylor series expansionof order two (Schmitt-Grohe, and Uribe (2004)). The semi-global approach in-dicates superior performance in accuracy and inherits global properties from theexact solution.

This paper contributes to a growing literature on using the perturbationtechnique for solving DSGE models. The perturbation methodology in eco-

1The algorithms incorporated in the widely-used software such as Dynare (and less availableTroll) find a stacked-time solution and are based on Newton’s method combined with sparse-matrix techniques (Adjemian, Bastani, Juillard, Karame, Mihoubi, Perendial, Pfeifer, Ratto,and Villemot (2011)).

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nomics has been advanced by Judd and co-authors as in Judd (1998); Gaspar,and Judd (1997); Judd, and Guu (1997). Jin, and Judd (2002) give a theoret-ical basis for using perturbation methods in DSGE modeling; namely, applyingthe implicit function theorem, they prove that the perturbed rational expecta-tions solution continuously depends on a parameter and therefore tends to thedeterministic solution as the parameter tends to zero.

Almost all of the literature is concerned with the approximations aroundthe steady state as in Collard, and Juillard (2001); Schmitt-Grohe, and Uribe(2004); Kim, Kim, Schaumburg, and Sims (2008); Gomme, and Klein (2011).Lombardo (2010) uses series expansion in powers of σ to find approximationsto the exact solution recursively. Borovicka, and Hansen (2013) employ Lom-bardo’s approach to construct shock-exposure and shock-price elasticities, whichare asset-pricing counterparts to impulse response functions. This approach hassome similarity with that employed in the current paper. However, both papersapply the expansion only around the deterministic steady state, therefore thesolution obtained remains local. Lombardo’s approach can be treated as a spe-cial case of the method proposed in this study, namely a deterministic solutionaround which the expansion is used is only the steady state.

Judd (1998, Chapter 13) outlines how to apply perturbations around theknown entire solution, which is not necessarily the steady state. He considers thesimple continuous and discrete-time stochastic growth models in the dynamicprogramming framework. This paper develops a rigorous approach to constructsolutions to DSGE models in general form by using the perturbation methodaround a global deterministic path.

The rest of the paper is organized as follows. The next section presents themodel. Section 3 provides a detailed exposition of series expansions for DSGEmodels. In Section 4 we transform the model into a convenient form to dealwith. Section 5 presents the method for solving rational expectations models fortime-varying parameters. The proposed method is applied to an asset pricingmodel in Section 6, where it also compared with the local perturbation methodin terms of accuracy. Conclusions are presented in Section 7.

2 The Model

DSGE models usually have the form

Etf(yt+1, yt, xt+1, xt, zt+1, zt) = 0, (2.1)

zt+1 = Λzt + σεt+1, εt+1 ∼ N(0,Ω), (2.2)

where Et denotes the conditional expectations operator, xt is an nx × 1 vectorcontaining the t-period endogenous state variables; yt is an ny × 1 vector con-taining the t-period endogenous variables that are not state variables; zt is annz × 1 vector containing the t-period exogenous state variables; εt is the vectorwith the corresponding innovations; σΩ is the nz × nz covariance matrix of theinnovations; f maps Rny ×R

ny ×Rnx ×R

nx ×Rnz ×R

nz into Rny ×R

nx and is

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assumed to be sufficiently smooth. The scalar σ (σ > 0) is a scaling parameterfor the disturbance terms εt. We assume that all mixed moments of εt are finite.All eigenvalues of the matrix Λ have modulus less than one.

The solution to (2.1) and (2.2) is of the form:

yt = h(xt, zt), (2.3)

where h maps Rnx ×Rnz into R

ny . Another way of stating the problem to solveis to say: for a given initial condition (x0, z0) find the initial condition y0 forthe vector y such that the solution (xt, yt) to (2.1)–(2.2) will be bounded for allt > 0.

3 Series Expansion

In this section we shall follow the perturbation methodology (see, for example,Holmes (2013)) to derive an approximate solution to the model (2.1)–(2.2). Forsmall σ, we assume that the solution has a particular form of expansions

yt =

∞∑

n=0

σny(n)(xt, zt) (3.1)

xt =

∞∑

n=0

σnx(n)t , (3.2)

where y(i)(xt, zt) and x(i)t , i = 0, 1, 2, . . . , are the i-order of approximation to

the solution (2.3) and the variable xt, respectively. The exogenous process ztcan also easily be represented in the form of expansion in σ

zt = z(0)t + σz

(1)t . (3.3)

Indeed, plugging (3.3) into (2.2) gives

zt+1 = z(0)t+1 + σz

(1)t+1 = Λ(z

(0)t + σz

(1)t ) + σεt+1.

Collecting the terms of like powers of σ and equating them to zero, we get

z(0)t+1 = Λz

(0)t , (3.4)

z(1)t+1 = Λz

(1)t + εt+1. (3.5)

Since the expansion (3.3) must be valid for all σ at the initial time t = 0, theinitial conditions are

z(0)0 = z0 and z

(1)0 = 0. (3.6)

Note that the arguments of the functions y(i) are expansions in powers of σ.Substituting (3.2) and (3.3) into (3.1) yields

4

yt =

∞∑

i=0

σiy(i)

∞∑

j=0

σjx(j)t , z

(0)t + σz

(1)t

. (3.7)

Expanding yt for small σ and collecting the terms of like powers, we have

yt =∞∑

n=0

σny∗(n)(x(0)t , x

(1)t , ..., x

(n)t , z

(0)t , z

(1)t ), (3.8)

wherey∗(0)(x

(0)t , z

(0)t ) = y(0)(x

(0)t , z

(0)t ),

y∗(1)(x(0)t , x

(1)t , z

(0)t , z

(1)t ) = y(1)(x

(0)t , z

(0)t ) + y

(0)1,0;tx

(1)t + y

(0)0,1;tz

(1)t ,

and

y∗(n)(x(0)t , x

(1)t , ..., x

(n)t , z

(0)t , z

(1)t ) = y(n)(x

(0)t , z

(0)t ) + y

(0)1,0;tx

(n)t + pn,t, (3.9)

where the mapping pn,t = pn(x(0)t , x

(1)t , . . . , x

(n−1)t , z

(0)t , z

(1)t ) has arguments

with superscript less than n and is defined as

pn,t =

n∑

l=0

1

l!

n−l∑

j=0

n−j−l∑

k=1

1

k!y(j)k,l;t

∑

i1+i2+···+ik=n−j−l

(

n− j − l

i1; i2; ...; ik

)

x(i1)t , x

(i2)t , ..., x

(ik)t ,

(

z(1)t

)l

Here y(j)k,l;t denotes the mixed partial derivative of y(j) of order k and l with re-

spect to xt and zt, respectively, at the point (x(0)t , z

(0)t ), and

(

z(1)t

)l

= (z(1)t , . . . , z

(1)t )

(l times). In other words, y(j)k,l;t is a (k + l)-multilinear mapping (see, for exam-

ple, Abraham, Marsden, and Ratiu (2001, p. 55)) depending on (x(0)t , z

(0)t ) (and

hence on t). Substituting (3.9) into (3.8), we can rewrite (3.8) as

yt =

∞∑

n=0

σn[

y(n)(x(0)t , z

(0)t ) + y

(0)1,0;tx

(n)t + pn,t

]

. (3.10)

Then substituting (3.2), (3.3) and (3.10) into (2.1), collecting the terms of likepowers of σ and setting their coefficients to zero, we have

Coefficient of σ0

f(

y(0)(x(0)t+1, z

(0)t+1), y

(0)(x(0)t , z

(0)t ), x

(0)t+1, x

(0)t , z

(0)t+1, z

(0)t

)

= 0, (3.11)

The requirement that (3.2) and (3.3) must hold for all arbitrary small σ impliesthat the initial conditions for (3.11) are

z(0)0 = z0 and x

(0)0 = x0. (3.12)

The terminal condition is the steady state. The system of equations (3.4) and(3.11) is a deterministic model since it corresponds to the model (2.1) and

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(2.2), where all shocks vanish. The deterministic model (3.4) and (3.11) withthe initial conditions (3.12) can be solved globally by a number of effectivealgorithms, for example the extended path method ( Fair, and Taylor (1983)) ora Newton-like method (for example, Juillard (1996)). As this study is primarilyconcerned with stochastic models, in what follows we suppose that the solution

(x(0)t , y(0)(x

(0)t , z

(0)t )) for t > 0 to the deterministic model is already known.

Coefficient of σn, n > 0

Et

f1,t+1 · y(n)t+1 + f2,t+1 · y

(n)t +

[

f1,t+1 · y(0)1,0;t+1 + f3,t+1

]

x(n)t+1

+[

f2,t+1 · y(0)1,0;t + f4,t+1

]

x(n)t + η

(n)t+1

= 0,(3.13)

where y(n)t = y(n)(x

(0)t , z

(0)t ). The requirement that (3.2) must hold for all

arbitrary small σ implies that the initial condition for (3.13) is

x(n)0 = 0. (3.14)

The matrices

fi,t+1 = fi

(

y(0)(x(0)t+1, z

(0)t+1), y

(0)(x(0)t , z

(0)t ), x

(0)t+1, x

(0)t , z

(0)t+1, z

(0)t

)

, i = 1, . . . , 6,

are the Jacobian matrices of the mapping f with respect to yt+1, yt, xt+1, xt,zt+1, and zt, respectively, at the point(

y(0)(x(0)t+1, z

(0)t+1), y

(0)(x(0)t , z

(0)t ), x

(0)t+1, x

(0)t , z

(0)t+1, z

(0)t

)

.

The mapping Etη(n)t is of the form:

Etη(n)t+1 = Etη

(n)(

x(0)t+1, x

(0)t , . . . , x

(n−1)t+1 , x

(n−1)t , z

(0)t+1, z

(0)t , z

(1)t+1, z

(1)t

)

,

where η(n) is some mapping for which the set of arguments includes only quan-

tities of order less than n. The vector z(1)t+1 enters the expectations Etη

(n)t+1 in

the form of the mixed moments of order n or less. The subscript t+ 1 in fi,t+1

and η(n)t+1 reflects their dependence on t+ 1 through x

(0)t+1 and z

(0)t+1.

The expectation Etη(n)t+1 is bounded if all mixed moments of z

(1)t+1 are bounded

up to order n and the vectors(

y(0)t+1, y

(0)t , x

(0)t+1, x

(0)t , . . . , y

(n−1)t+1 , y

(n−1)t , x

(n−1)t+1 , x

(n−1)t , z

(0)t+1, z

(0)t , z

(1)t+1, z

(1)t

)

are bounded for all t ≥ 0.Equation (3.13) with the initial conditions (3.14) is a linear rational expecta-

tions model with time-varying coefficients. To solve the problem (3.13)–(3.14) is

equivalent to find a bounded solution (x(n)t , y

(n)t ) for t > 0 under the assumption

that the bounded solutions to the problems of all orders less than n are alreadyknown. It is worth noting that the homogeneous part of (3.13) is the same for

all n > 0 and the difference is only in the non-homogeneous terms Etη(n)t+1. In

Section 5 we present a method for solving such types of model and prove theconvergence of the solutions implied by the method to the exact solution. Inthe next section we transform equation (3.13) in a more convenient form to dealwith.

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4 Transformation of the Model

Define the deterministic steady state as vectors (y, x, 0) such that

f(y, y, x, x, 0, 0) = 0. (4.1)

We can represent fi,t+1 in (3.13) as fi,t+1 = fi + fi,t+1, i = 1, . . . , 6, wherefi = fi(y, y, x, x, 0, 0) are the Jacobian matrices of the mapping f with respectto yt+1, yt, xt+1, xt, zt+1, and zt, respectively, at the steady state, and

fi,t+1 = fi,t+1(y(0)t+1, y

(0)t , x

(0)t+1, x

(0)t , z

(0)t+1, z

(0)t )− fi(y, y, x, x, 0, 0). (4.2)

Note also that fi,t+1 → 0 as t → ∞, because a deterministic solution must tendto the deterministic steady state as t tends to infinity. Consequently, fi,t+1 canbe thought of as a perturbation of fi.

To shorten notation, further on we omit the superscript (n) when no confu-sion can arise. Therefore Equations (3.13) can be written in the vector form

Φt+1Et

[

xt+1

yt+1

]

= Λt+1

[

xt

yt

]

+ Etηt+1, (4.3)

where Φt =[

f3 + f3,t, f1 + f1,t

]

and Λt =[

f4 + f4,t, f2 + f2,t

]

. We assume

that the matrices Φt are invertible for all t ≥ 0. This assumption holds if, forexample, the Jacobian [f3, f1]

−1at the steady state is invertible2 and the terms

f1,t and f3,t are small enough for all t ≥ 0. Pre-multiplying (4.3) by Φ−1t+1, we

get

Et

[

xt+1

yt+1

]

= L

[

xt

yt

]

+Mt+1

[

xt

yt

]

+Φ−1t+1Etηt+1, (4.4)

where L = [f3, f1]−1

[f4, f2] and

Mt+1 =[

f3 + f3,t+1, f1 + f1,t+1

]−1 [

f4 + f4,t+1, f2 + f2,t+1

]

−[f3, f1]−1

[f4, f2] .

Notice that limt→∞Mt = 0. As in the case of rational expectations modelswith constant parameters it is convenient to transform (4.4) using the spectralproperty of L. Namely, the matrix L is transformed into a block-diagonal oneusing the block-diagonal Schur factorization3

L = ZPZ−1, (4.5)

where

P =

[

A 00 B

]

; (4.6)

2This assumption is made for ease of exposition. If [f3, f1] is a singular matrix, thenfurther on we must use a generalized Schur decomposition for which derivations remain valid,but become more complicated.

3The function bdschur of Matlab Control System Toolbox performs this factorization.

7

where A and B are quasi upper-triangular matrices with eigenvalues larger andsmaller than one (in modulus), respectively; and Z is an invertible matrix4. Wealso impose the conventional Blanchard-Kan condition (Blanchard, and Kahn(1980)) on the dimension of the unstable subspace, i.e., dim(B) = ny.

After introducing the auxiliary variables

[st, ut]′ = Z−1[xt, yt]

′ (4.7)

and pre-multiplying (4.4) by Z−1, we have

Etst+1 = Ast +Q11,t+1st +Q12,t+1ut +Ψ1t+1Etηt+1, (4.8)

Etut+1 = But +Q21,t+1st +Q22,t+1ut +Ψ2t+1Etηt+1, (4.9)

where [Ψ1,t+1,Ψ2,t+1] = ZΦ−1t+1 and

[

Q11,t+1 Q12,t+1

Q21,t+1 Q22,t+1

]

= ZMt+1Z−1. (4.10)

System (4.8)-(4.9) is a linear rational expectations model with time-varyingparameters, thus we cannot apply the approaches used in the case of modelswith constant parameters (Blanchard, and Kahn (1980); Anderson and Moor(1985); Sims (2001); Uhlig (1999), etc.). In Subsection 5.2 we devevop a methodfor solving this type of models.

5 Solving the Rational Expectations Model with

Time-Varying Parameters

5.1 Notation

This subsection introduces some notation that will be necessary further on. By|·| denote the Euclidean norm in R

n. The induced norm for a real matrix D isdefined by

‖D‖ = sup|s|=1

| Ds|.

The matrix Z in (4.5) can be chosen in such a way that

‖A‖ < α+ γ < 1 and ‖B−1‖ < β + γ < 1, (5.1)

where α and β are the largest eigenvalues (in modulus) of the matrices A andB−1, respectively, and γ is arbitrarily small. This follows from the same argu-ments as in Hartmann (1982, §IV 9), where it is done for the Jordan matrix

decomposition. Note also that ‖B‖−1

< 1 for sufficiently small γ. Let

Bt = B +Q22,t

4A simple generalized Schur factorization is also possible to employ here, but at the costof more complicated derivations.

8

and At = A+Q11,t.(5.2)By definition, put

a = supt=0,1,...

‖At‖ , b = supt=0,1,...

∥

∥B−1t

∥

∥ , (5.3)

c = supt=0,1,...

‖Q12,t‖ , d = supt=0,1,...

‖Q21,t‖ . (5.4)

Here and in what follows we assume that all the matrices Bt, t = 0, 1, . . . ,are invertible. The numbers a, b, c and d depend on the initial conditions

(x(0)0 , z

(0)0 ). From the definitions of At, A, Bt, B, Q12,t and Q21,t and the

condition limt→∞(x(0)t , z

(0)t ) = (x, 0), it follows that

limt→∞

c(x(0)t , z

(0)t ) = 0, lim

t→∞d(x

(0)t , z

(0)t ) = 0, (5.5)

limt→∞

a(x(0)t , z

(0)t ) = ‖A‖ < 1, lim

t→∞b(x

(0)t , z

(0)t ) =

∥

∥B−1∥

∥ < 1.

This means that c and d can be arbitrary small and

a < 1 and b < 1 (5.6)

by choosing (x(0)0 , z

(0)0 ) close enough to the steady state.

5.2 Solving the transformed system (4.8)–(4.9)

Taking into account notation (5.1), we can rewrite (4.8)–(4.9) in the form

Etst+1 = At+1st +Q12,t+1ut +Ψ1,t+1Etηt+1, (5.7)

Etut+1 = Bt+1ut +Q21,t+1st +Ψ2,t+1Etηt+1. (5.8)

In this subsection we construct a bounded solution to (5.7)–(5.8) for t ≥ 0 withan arbitrary initial condition s0 ∈ R

nx and find under which conditions thissolution exists. For this purpose, we first start with solving a finite-horizonmodel with a fixed terminal condition using backward recursion. Then, weprove the convergence of the obtained finite-horizon solutions to a boundedinfinite-horizon one as the terminal time T tends to infinity.

Fix a horizon T > 0. Using the invertibility of BT=1 and solving Equa-tion (5.8) backward, we can obtain uT as a linear function of sT , the terminalcondition ETuT+1 and the “exogenous” term Ψ2,T+1ET ηT+1

uT = −B−1T+1Q21,T+1sT −B−1

T+1Ψ2,T+1ET ηT+1 +B−1T+1ETuT+1.

Proceeding further with backward recursion, we shall obtain finite-horizon so-lutions for each t = 0, 1, 2, . . . , T. For doing this we need to define the followingrecurrent sequence of matrices:

KT,T−i−1 = L−1T+1,T−i (Q21,T−i +KT,T−iAT−i) , i = 0, 1, . . . , T, (5.9)

whereLT,T−i = BT−i +KT,T−iQ12,T−i, (5.10)

9

with the terminal conditionKT,T+1 = 0. In (5.9) and (5.10) the first subscript Tdefines the time horizon, while the second subscript defines all times between 0and T +1. Let uT,T−i, i = 0, 1, . . . , T, denote the (T − i)-time solution obtainedby backward recursion that starts at the time T .

Proposition 5.1. Suppose that the sequence of matrices (5.9) and (5.10) exists;then the solution to (5.7)–(5.8) has the following representation:

uT,T−i = −KT,T−isT−i + gT,i +

(

i+1∏

k=1

L−1T,T−i+k

)

ET−i (uT+1) , (5.11)

where i = 0, 1, . . . , T ; and

gT,i = −

i+1∑

j=1

j∏

k=1

L−1T,T−i+k(Ψ2,T−i+j +KT,T−i+jΨ1,T−i+j)ET−iηT−i+j . (5.12)

For the proof see Appendix A. The sequence of matrices (5.9) exists if allmatrices LT,T−i, i = 0, 1, . . . , T, are invertible. For this we need, in addition,some boundedness condition on the matrices B−1

T−iKT,T−i+1Q12,T−i.

Proposition 5.2. If for a, b, c and d from (5.3)–(5.4) the inequality

cd <1

4

(

1

b− a

)2

=

(

1− ab

2b

)2

(5.13)

holds, then

∥

∥B−1T−i

∥

∥ · ‖KT,T−i+1‖ · ‖Q12,T−i‖ < 1, i = 0, 1, 2, . . . T. (5.14)

For the proof see Appendix A.

Proposition 5.3. If inequality (5.14) holds, then the matrices LT,T−i, i =0, 1, 2. . . . , T , are invertible.

Proof. From (5.10) and the invertibility of BT−i it follows that

LT,T−i = BT−i

(

I +B−1T−iKT,T−iQ12,T−i

)

. (5.15)

The matrices LT,T−i are invertible if and only if the matrices(

I +B−1T−iKT,T−iQ12,T−i

)

are invertible. From the norm property and (5.14) we have∥

∥B−1T−iKT,T−i+1Q12,T−i

∥

∥ ≤∥

∥B−1T−i

∥

∥ · ‖KT,T−i+1‖ · ‖Q12,T−i‖ < 1.

Now the invertibility of(

I +B−1T−iKT,T−iQ12,T−i

)

follows from Golub, and VanLoan (1996, Lemma 2.3.3)

For i = T from (5.11) we have

uT,0 = −KT,0s0 + gT,T +

(

T+1∏

k=1

L−1T,k

)

E0 (uT+1) . (5.16)

10

This is a finite-horizon solution to the rational expectations model with time-varying coefficients (5.7)–(5.8) and with a given initial condition s0. What isleft is to show that the solution uT,0 of the form (5.16) converges to some limitas T → ∞.

Proposition 5.4. If inequality (5.13) holds, then the limit

limT→∞

KT,j = K∞,j for j = 0, 1, 2, . . .

exists in the matrix space defined in Subsection 5.1.

For the proof see Appendix A.

Proposition 5.5. If inequality (5.14) holds, then

limT→∞

T+1∏

k=1

L−1T,k = 0 (5.17)

and

limT→∞

gT,T = g∞, (5.18)

where g∞ is some vector in Rny .

Proof. From (5.10) and Proposition 5.4 it follows that

limT→∞

LT,k = Bk +K∞,kQ12,k = L∞,k.

Then the limit in (5.17) can be represented as

limT→∞

T+1∏

k=1

L−1T,k = lim

T→∞

T+1∏

k=1

L−1∞,k. (5.19)

Since K∞,k is bounded (it follows from formula (A.7) in Appendix A) and

limk→∞

Q12,k = 0, and limk→∞

B−1k = B−1,

we have limk→∞L−1∞,k = B−1. Therefore, if δ > 0 is arbitrary small, there is an

N = Nδ ∈ N such that

‖L−1∞,k‖ ≤ β + δ = ρ < 1, (5.20)

for k > N , where β is the largest eigenvalue (in modulus) of the matrix B−1.From this, the norm property and (5.19) we obtain

limT→∞

∥

∥

∥

∥

∥

T+1∏

k=1

L−1T,k

∥

∥

∥

∥

∥

≤ limT→∞

T+1∏

k=1

∥

∥

∥L−1∞,k

∥

∥

∥ ≤ limT→∞

C1ρT−K = 0,

where C1 is some constant. Therefore, (5.17) is proved.

11

By (5.20) the products in (5.12) decay exponentially with the factor ρ asj → ∞. From this and the boundedness of the terms KT,k, Ψ2,k, Ψ1,k andE0ηk, T ∈ N and k = 1, 2, . . . , T + 1, it follows that the series

gT,T = −

T+1∑

j=1

j∏

k=1

L−1T,k(Ψ2,j +KT,jΨ1,j)E0ηj .

converges to some g∞ as T → ∞.

From Proposition 5.4 and Proposition 5.5 it may be concluded that as Ttends to infinity Equation (5.16) takes the form:

u0 = −K∞,0s0 + g∞. (5.21)

Formula (5.21) gives us the unique bounded solution to the transformed rationalexpectation model with time-varying parameters (5.7)–(5.8). Note also that theproofs of Propositions 5.2–5.5 are based on inequality (5.13) that is a spectralgap condition for the unstable and stable parts of the system (5.7)–(5.8), and ina sense substitutes for the Blanchard-Kahn condition for rational expectationsmodels with time-varying parameters. It follows from (5.3)–(5.6) that inequality

(5.13) always holds if initial conditions (x(0)t , z

(0)t ) is close enough to the steady

state. Nonetheless, the condition (5.13) is not local by itself.

5.3 Restoring the original variables x(n)t and y

(n)t

Recall that we deal with the n-order problem (3.13)–(3.14), and now we putthe superscript (n) back in notation. To find the bounded solution in terms of

the original variables x(n)t and y

(n)t we need to obtain the initial values u

(n)0 and

s(n)0 that correspond to that of the problem (4.4), i.e. x

(n)0 = 0. From (4.7) and

(5.21) we have[

s(n)0

−K(n)∞,0s

(n)0 + g

(n)∞

]

= Z−1

[

0

y(n)0

]

,

where Z−1 is a matrix that is involved in the block-diagonal Schur factorization(4.5) and has the following block-decomposition:

Z−1 =

[

Z11 Z12

Z21 Z22

]

.

Hence

s(n)0 = Z12y

(n)0 , (5.22)

−K(n)∞,0s

(n)0 + g(n)∞ = Z22y

(n)0 . (5.23)

Substituting (5.22) into (5.23) and assuming that the matrix Z22+K(n)∞,0Z

12

is invertible, we get

y(n)0 = (Z22 +K

(n)∞,0Z

12)−1g(n)∞ . (5.24)

12

The left-hand side of (5.24) corresponds to y(n)(x0, z0) in (3.1). The dependence

of y(n)0 on (x0, z0) follows from K

(n)∞,0 and g

(n)∞ . Therefore, formula (5.24) deter-

mines the solution to the original rational expectations model with time-varying

parameters (3.13) and with the initial condition x(n)0 = 0. By assumption, the

solutions of lower order are already computed, thus the policy function approx-imation is of the form5

yt =

n∑

i=0

σiy(i)(xt, zt).

The matrix (Z22 + K∞,0Z12) is invertible if (i) the matrix Z22 is square and

invertible, and (ii) the norm of the matrix K∞,0 is small enough. The condition(i) corresponds to Proposition 1 of Blanchard, and Kahn (1980); at the same

time, the condition (ii) can be always attained if initial conditions (x(0)t , z

(0)t )

are close enough to the steady state, which follows from (A.6) and (A.7) ofAppendix A. Notice again that these conditions are not local by themselves.

If we are interested in finding dynamics, for example, impulse response func-

tions; then knowing y(n)0 and using (5.22)–(5.24) we can recover initial conditions

(s(n)0 , u

(n)0 ) in terms of the variables s(n) and u(n), solve equations (5.7)–(5.8)

with these initial conditions , and finally obtain the solution to (4.4), using thetransformation Z. This provides the solution to (4.4) in the form

x(n)t = Z11s

(n)t + Z12u

(n)t ,

y(n)t = Z21s

(n)t + Z22u

(n)t ,

where Zij , i = 1, 2, j = 1, 2, are blocks of the block-decomposition of the matrixZ.

6 An Asset Pricing Model

In this section we apply the presented method to an nonlinear asset pricingmodel proposed by Burnside (1998) and analyzed by Collard, and Juillard(2001). The representative agent maximizes the lifetime utility function

max

(

E0

∞∑

t=0

βtCθt

θ

)

subject toptet+1 + Ct = ptet + dtet,

where β > 0 is a subjective discount factor, θ < 1 and θ 6= 0, Ct denotesconsumption, pt is the price at date t of a unit of the asset, et represents units

5In fact, it is not hard to prove that in the case of symmetric distribution of εt for all odd

n the unique bounded solution is x(n)t

≡ 0 and y(n)t

≡ 0. We will show this for a simple assetpricing model in Section 6 for i = 1.

13

of a single asset held at the beginning of period t, and dt is dividends per assetin period t. The growth of rate of the dividends follows an AR(1) process

xt = (1− ρ) x + ρxt−1 + σεt+1, (6.1)

where xt = ln(dt/dt−1), and εt+1 ∼ NIID(0, 1). The first order condition andmarket clearing yields the equilibrium condition

yt = βEt [exp(θxt+1) (1 + yt+1)] , (6.2)

where yt = pt/dt is the price-dividend ratio. This equation has an exact solutionof the form (Burnside (1998))

yt =

∞∑

i=1

βi exp [ai + bi(xt − x)] , (6.3)

where

ai = θxi+1

2

(

θσ

1− ρ

)2 [

i−2ρ(1− ρi)

1− ρ+

ρ2(1− ρ2i)

1− ρ2

]

(6.4)

and

bi =θρ(1− ρi)

1− ρ.

It follows from (6.2) that the deterministic steady state of the economy is

y =β exp(θx)

1− β exp(θx).

6.1 Solution

We now obtain a solution to the system (6.1)–(6.2) as an expansion in powersof the parameter σ using the second-order approximation method developed inSections 3–5. Specifically, we are seeking for the solution of the form:

yt = y(0)(xt) + σy(1)(xt) + σ2y(2)(xt) (6.5)

xt = x(0)t + σx

(1)t . (6.6)

Substituting (6.6) into (6.1) and collecting the terms containing σ0 and σ1, weobtain the representation (6.6) for xt

x(0)t+1 = (1− ρ) x + ρx

(0)t (6.7)

x(1)t+1 = ρx

(1)t + εt+1. (6.8)

Since the expansion (6.6) must be valid for all σ at the initial time t = 0, theinitial conditions are

x(0)0 = x0 and x

(1)0 = 0. (6.9)

14

Substituting (6.5) and (6.6) into (6.2), then collecting the terms of like pow-ers of σ and setting the coefficients of like powers of σ to zero, we have (fordetails see Appendix B)

Coefficient of σ0

y(0)t = β exp(θx

(0)t+1)(1 + y

(0)t+1), (6.10)

x(0)t+1 = ρx

(0)t . (6.11)

Coefficient of σ1

y(0)1;t x

(1)t + y

(1)t =

+ exp(θx(0)t+1)βEt

[

θx(1)t+1(1 + y

(0)t+1) + y

(0)1;t+1x

(1)t+1 + y

(1)t+1

]

,(6.12)

x(1)t+1 = ρx

(1)t + εt+1.

Coefficient of σ2

y(2)t = −y11;tx

(1)t − 1

2y(0)2;t

(

x(1)t

)2

+ 12β[

θ2(

1 + y(0)t+1

)

+ 2θy(0)1;t+1 + y

(0)2,t+1

]

exp(θx(0)t+1)Et

(

x(1)t+1

)2

+β exp(θx(0)t+1)Et

[

y(1)t+1 + x

(1)t+1

(

y(1)1;t+1 + θy

(1)t+1

)

+ Et

(

y(2)t+1

)]

,

(6.13)

where y(i)j;t , i = 0, 1, j = 1, 2, are derivatives of y(i) of order j at the point x

(0)t .

Here and further on, for the simplicity of notation, we write y(i)t instead of

y(i)(x(0)t ), i = 0, 1, 2.

The system (6.10)-(6.1) is a deterministic model. Its solution can be obtainedby taking σ = 0 in (6.3) and (6.4)

y(0)t =

∞∑

i=1

βi exp

θ

[

xi+ρ(1− ρi)

1− ρ(xt − x)

]

, (6.14)

For the first order approximation we can rewrite (6.12) in the form

y(0)1;t x

(1)t + y

(1)t = β exp(θx

(0)t+1)

[

θ(1 + y(0)t+1) + y

(0)1;t+1

]

Etx(1)t+1

+ β exp(θx(0)t+1)Ety

(1)t+1

(6.15)

Under the assumption that y(0)t and x

(0)t are known for t ≥ 0, Equations (6.15)

and (6.8) constitute a forward looking model. Since x(1)0 = 0, from (6.8) we

have E0x(1)t = 0 for t > 0. It is easily shown that the only bounded solution of

(6.15) is y(1)t ≡ 0 for t ≥ 0.

Equation (6.13) is a linear forward-looking equation with time varying de-terministic coefficients. This equation can be solved by the backward recursion

15

considered in Section 5. Taking into account that the initial value of x(1)t is zero,

it can be easily checked that the solution of (6.13) has the form

y(2)t =

1

2

∞∑

n=1

βn exp[

θ(

x(0)t+1 + x

(0)t+2 + · · ·+ x

(0)t+n

)]

·[

θ2(

1 + y(0)t+n

)

+ 2θy(0)1;t+n

]

· Et

(

x(1)t+n

)2

(6.16)

Here y(0)1;t+n can be obtained by differentiating (6.3) with respect to xt and is

given by

y(0)1;t =

∞∑

i=1

βi ρ(1− ρi)

1− ρexp

θ

[

xi+ρ(1− ρi)

1− ρ(xt − x)

]

.

From (6.8) and (6.9) we have the moving-average representation for x(1)t+1:

x(1)t+n = εt+n + ρεt+n−1 + ...+ ρn−1εt+1.

Since the sequence of innovations εt, t > 0, is independent it follows that

Et

(

x(1)t+n

)2

= Et

(

εt+n + ρεt+n−1 + · · ·+ ρn−1εt+1

)2

= 1 + ρ2 + · · ·+ ρ2(n−1) =1− ρ2n

1− ρ2.

(6.17)

From (6.7) we have

x(0)t+1 + x

(0)t+2 + · · ·+ x

(0)t+n = x+ ρ(x

(0)t − x) + x+ ρ2(x

(0)t − x)

+x+ ρn(x(0)t − x) = nx+ ρ(1−ρn)

1−ρ(x

(0)t − x).

(6.18)

Finally, inserting (6.17) and (6.18) into (6.16) gives

y(2)t =

θ2

2

∞∑

n=1

βn 1− ρ2n

1− ρ2exp

θ[

nx+ θρ(1− ρn)

1− ρ(x

(0)t − x)

]

[

θ2(1 + y(0)t+n) + 2θy

(0)1;t+n

]

.

To summarize, we find the policy function approximation in the form

y(x) = y(0)t (x) + σ2y

(2)t (x).

The solutions for the higher orders y(i)t (x), i > 2, can be obtained in much the

same way as for y(2)t (x). Note also that it is easily shown that for all odd i the

unique bounded solution is y(i)t ≡ 0.

6.2 Accuracy Check

This subsection compares the accuracy of the second order of the presentedmethod with the local Taylor series expansions of order 2 (Schmitt-Grohe, and

16

Uribe (2004)). The following three criteria are used to check the accuracy ofthe approximation methods:

E0,∞ = 100 ·maxi

∣

∣

∣

∣

y(xi)− y(xi)

y(xi)

∣

∣

∣

∣

,

E1,∞ = 100 ·maxi

∣

∣

∣

∣

∆y(xi)−∆y(xi)

∆y(xi)

∣

∣

∣

∣

,

E2,∞ = 100 ·maxi

∣

∣

∣

∣

∆2y(xi)−∆2y(xi)

∆2y(xi)

∣

∣

∣

∣

,

where y(xi) denotes the closed-form solution, y(xi) is an approximation of thetrue solution by the method under study, ∆y(xi) = y(xi) − y(xi − ∆x) and∆x = xi − xi−1 are the first difference of y and x, respectively, ∆2y(xi) isthe second difference of y, i.e ∆2y(xi) = ∆y(xi) − ∆y(xi−1). The criterionE0,∞ is the maximal relative error made using an approximation rather thanthe true solution. The criteria E1,∞ and E2,∞ capture the accuracy of thecharacteristics of the shape of an approximate policy function, namely the slopeand convexity, by comparing the maximal relative first and second differencesof an approximate and the closed-form solutions. All criteria are evaluated overthe interval xi ∈ [x−∆ ·σx, x+∆ ·σx], where σx is the unconditional volatilityof the process xt and ∆ = 5.

The parameterization follows Collard, and Juillard (2001), where the bench-mark parameterization is chosen as in Mehra, and Prescott (1985). We thereforeset the mean of the rate of growth of dividend to x = 0.0179, its persistence toρ = -0.139 and the volatility of the innovations to σ = 0.0348. The parameterθ was set to −1.5 and β to 0.95. We investigate the implications of larger cur-vature of the utility function, higher volatility and more persistence in the rateof growth of dividends in terms of accuracy.

Table 1 shows that the maximal relative error for the benchmark parameteri-zation is three times larger for the approximation of the Taylor series expansionthan for the semi-global method, however the errors for both these methodsare very small - 0.06% and 0.02%, respectively. The increase in the condi-tional volatility of the rate of growth of dividends to σ = 0.1 yields the higherapproximation errors of 2% and 1% for the local Taylor series expansion andsemi-global method, respectively. Increasing the curvature of the utility func-tion to θ = −10 yields the maximal approximation error 8.4% for the Taylorseries expansion approximation and about two times smaller for the semi-globalmethod.

The semi-global method becomes considerably more accurate than the localTaylor series expansion if the persistence of the exogenous process increases.For the parameterization ρ = 0.5 and σ = 0.03 the proposed method gives themaximal relative approximation error six times smaller than the local Taylor se-ries expansion. Increasing the persistence to ρ = 0.9 yields the maximal relativeapproximation error of 193% for the local Taylor series expansion and 9% forthe semi-global method. This effect is more pronounced for the criteria E1,∞

17

Table 1: The relative errors of the approximate solutions

Criterion E0,∞ E1,∞ E2,∞

Model SGa P2b SG P2 SG P2ParameterizationBenchmark 0.02 0.06 0.02 1.47 0.02 4.53θ = −10 4.75 8.39 4.66 25.0 4.56 37.6σ = 0.1 1.30 2.23 1.29 12.0 1.28 19.3ρ = 0.5 0.26 1.56 0.28 8.72 0.30 26.6ρ = 0.5, θ = −5 10.3 27.8 11.0 69.4 11.6 71.3ρ = 0.9 9.30 193 11.3 392 12.8 360

a The semi-global method of order two

b The local Taylor series perturbation method of order two(Schmitt-Grohe, and Uribe (2004))

and E2,∞. Furthermore, for any parameterization the semi-global approxima-tion gives at least 5 times more accurate solution in the metrics E1,∞ and 9times in the metrics E2,∞ than the local Taylor series expansion.

Figure 1 shows the policy functions for the high persistence case, ρ = 0.9,and indicates that the semi-global method traces globally the pattern of thetrue policy function much better than the local Taylor series expansion. More-over, from Figure 1 we can also see another undesirable property of the thelocal Taylor series expansion, namely this method can provide impulse responsefunctions with a wrong sign. Indeed, the steady state value of yt is y = 12.3.After a positive shock the true impulse response function is negative, whereasthe impulse response function implied by the local perturbation method is pos-itive, if the shock is large enough. Note also that the solution produced bythe semi-global method is indistinguishable from the true solution for positiveshocks (the bottom right corner of the Figure 1).

7 Conclusion

This study proposes a method based on a perturbation around a deterministicpath for constructing approximate solutions to DSGE models. The solutionsobtained are global in the state space whenever so is the deterministic solution.As by product, an approach to solve linear rational expectations models withdeterministic time-varying parameters is developed. This approach might bevaluable in itself, for example, it can be used to solve Markov-Switching DSGEmodels. All results are obtained for DSGE models in general form and provedrigorously.

The advantage of using the local perturbation methods lies in the fact thatthey can deal with medium and large-size models for reasonable computationaltime. However, this methods are intrinsically local as they employ perturba-tions around the steady state. Whereas the global methods used in DSGE

18

Figure 1: Comparison of the policy functions for ρ = 0.9.

modeling, such as projection and stochastic simulations, suffer from the curseof dimensionality, i.e. they can handle only small-dimension models. The pro-posed approach has a potential to solve high-dimensional models as it sharessome preferable properties with the local perturbation methods. Namely, thecomputational gain may come from calculation of conditional expectations. Tocompute the conditional expectations using the semi-global method all that weneed is to know the moments of the distribution up to the order of approxi-mation, while the use of the global methods mentioned above involves eitherstochastic simulations or quadratures. The former is time consuming, the latercan deal with only low-order integrals. The practical implementation of theapproach to larger size models the author leaves for future research.

A Proofs for Section 5

PROOF OF PROPOSITION 5.1: The proof is by induction on i. Suppose thati = 0. For the time T from (5.8) we have

ETuT+1 = BT+1uT +Q21,T+1sT +Ψ2,T+1ET ηT+1.

As BT+1 is invertible, we have

uT,T = −KT,T sT − gT,0 + L−1T,T,TETuT+1,

where KT,T = B−1T+1Q21,T+1; gT,0 = −B−1

T+1Ψ2,T+1ET ηT+1 and L−1T,T+1 =

B−1T+1. From (5.9), (5.10) and (5.12) it follows that the inductive assumption is

19

proved for i = 0. Assuming that (5.11) holds for i > 0, we will prove it for i+1.To this end, consider Equation (5.8) for the time t = T − i− 1. As the matrixBT−i is invertible, we obtain

uT,T−i−1 = −B−1T−iQ21,T−isT−i−1−B−1

T−iΨ2,T−iET−i−1ηT−i+B−1T−iET−i−1uT,T−i.

Substituting the induction assumption (5.11) for uT,T−i yields

uT,T−i−1 = −B−1T−iQ21,T−isT−i−1 −B−1

T−iΨ2,T−iET−i−1ηT−i

+B−1T−iET−i−1

[

−KT,T−isT−i + gT,i +(

∏i+1k=1 L

−1T,T−i+k

)

ET−i (uT+1)]

.

Substituting (5.7) for ET−i−1(sT−i) and using the law of iterated expectationsgives

uT,T−i−1 = −B−1T−iQ21,T−isT−i−1 −B−1

T−iΨ2,T−iET−i−1ηT−i +B−1T−igT,i

+B−1T−i

(

∏i+1k=1 L

−1T,T−i+k

)

ET−i−1 (uT+1)

+B−1T−i [−KT,T−i (AT−isT−i−1 +Q12,T−iuT,T−i−1 +Ψ1,T−iET−i−1ηT−i)] .

Collecting the terms with uT,T−i−1, sT−i−1 and ηT−i, we get

(

I +B−1T−iKT,T−iQ12,T−i

)

uT,T−i−1 = −B−1T−i

[

(Q21,T−i +KT,T−iAT−i)sT−i−1

+(Ψ2,T−i +KT,T−iΨ1,T−i)ET−i−1ηT−i + gT,i +(

∏i+1k=1 L

−1T,T−i+k

)

ET−i−1 (uT+1)]

Suppose for the moment that the matrix ZT,T−i = I + B−1T−iKT,T−iQ12,T−i is

invertible. Pre-multiplying the last equation by Z−1T,T−i, we obtain

uT,T−i−1 = −Z−1T,T−iB

−1T−i

[

(Q21,T−i +KT,T−iAT−i)sT−i−1

+(Ψ2,T−i +KT,T−iΨ1,T−i)ET−i−1ηT−i + gT,i

+(

∏i+1k=1 L

−1T,T−i+k

)

ET−i−1 (uT+1)]

.

Note that LT,T−i = BT−iZT,T−i; then using the definition of KT,T−i−1 (5.9),we see that

uT,T−i−1 = −KT,T−i−1sT−i−1

−L−1T,T−i (Ψ2,T−i +KT,T−iΨ1,T−i)ET−i−1ηT−i

+L−1T,T−igT,i + L−1

T,T−i

(

∏i+1k=1 L

−1T,T−i+k

)

ET−i−1 (uT+1) .

(A.1)

Using the definition of gT,i and LT−i,T−i+j ((5.10) and (5.12)), we deduce that

gT,i+1 = −L−1T,T−i (Ψ2,T−i +KT,T−iΨ1,T−i)ET−i−1ηT−i + L−1

T,T−igT,i. (A.2)

From (A.1) and (A.2) it follows that

uT,T−i−1 = −KT,T−i−1sT−i−1 + gT,i+1 +

(

i+2∏

k=1

L−1T,T−i−1+k

)

ET−i−1 (uT+1) .

20

This proves the proposition.PROOF OF PROPOSITION 5.2: We begin by rewriting (5.9) as

(BT−i +KT,T−iQ12,T−i)KT,T−(i+1) = (Q21,T−i +KT,T−iAT−i) .

Rearranging terms, we have

KT,T−(i+1) = B−1T−i · (Q21,T−i +KT,T−iAT−i)

−B−1T−iKT,T−iQ12,T−iKT,T−(i+1).

(A.3)

Taking the norms and using the norm properties gives

∥

∥KT,T−(i+1)

∥

∥ ≤∥

∥B−1T−i

∥

∥ · ‖Q21,T−i‖+∥

∥B−1T−i

∥

∥ · ‖KT,T−i‖ · ‖AT−i‖

+∥

∥B−1T−i

∥

∥ · ‖KT,T−i‖ · ‖Q12,T−i‖ ·∥

∥KT,T−(i+1)

∥

∥ .

Rearranging terms, we get

∥

∥KT,T−(i+1)

∥

∥ ≤

∥

∥B−1T−i

∥

∥ · ‖Q21,T−i‖+∥

∥B−1T−i

∥

∥ · ‖KT,T−i‖ · ‖AT−i‖

1−∥

∥B−1T−i

∥

∥ · ‖KT,T−i‖ · ‖Q12,T−i‖. (A.4)

Inequality (A.4) is a difference inequality with respect to ‖KT,T−i‖, i =0, 1, . . . , T, and with time-varying coefficients ‖AT−i‖,

∥

∥B−1T−i

∥

∥, ‖Q12,T−i‖ and‖Q21,T−i‖. In (A.4) we assume that

1−∥

∥B−1T−i

∥

∥ · ‖KT,T−i‖ · ‖Q12,T−i‖ 6= 0.This is obviously true if ‖KT,T−i‖ = 0. We shall show that if the initial

condition ‖KT,T+1‖ = 0, then(

1−∥

∥B−1T−i

∥

∥ · ‖KT,T−i‖ · ‖Q12,T−i‖)

> 0, i =1, 2, . . . , T. Indeed, consider the difference equation:

si+1 =bd+ basi(1− bcsi)

. (A.5)

Lemma A.1. If inequality (5.13) holds, then the difference equation (A.5) hastwo fixed points

s∗1 =2bd

1− ba+√

(1− ba)2 − 4b2cd, (A.6)

s∗2 =1− ba+

√

(1− ba)2 − 4b2cd

2bc,

where s∗1 is a stable fixed point whereas s∗2 is an unstable one. Moreover, under

the initial condition s0 = 0 the solution si, i = 1, 2, . . . , is an increasing sequence

and converges to s∗1.

The lemma can be proved by direct calculation. From (5.4)–(5.3) the valuesa, b, c and d majorize ‖AT−i‖ ,

∥

∥B−1T−i

∥

∥, ‖Q12,T−i‖ and ‖Q21,T−i‖, respectively.If we consider Equation (A.2) and inequality (A.5) as initial value problemswith the initial conditions ‖KT,T+1‖ = 0 and s0 = 0, then their solutionsobviously satisfy the inequality ‖KT,T−i‖ ≤ si+1, i = 1, 2, . . . , T . In other

21

words, ‖KT,T−i‖ is majorized by si. From the last inequality and Lemma A.1it may be concluded that

‖KT,T−i‖ ≤ s∗1, i = 0, 1, 2, . . . , T, T ∈ N. (A.7)

From (A.6), (A.7) and (5.4) it follows that

∥

∥B−1T−i

∥

∥ · ‖KT,T−i‖ · ‖Q12,T−i‖ ≤2b2dc

1− ba+√

(1− ba)2 − 4b2cd. (A.8)

From (5.13) we see that 2b2dc < (1 − ab)2/2. Substituting this inequality into(A.8) gives

∥

∥B−1T−i

∥

∥ · ‖KT,T−i‖ · ‖Q12,T−i‖ ≤(1− ba)

2

2(1− ba+√

(1− ba)2 − 4b2cd)

<(1− ba)

2

2(1− ba)=

1− ba

2< 1,

(A.9)

where the last inequality follows from (5.6). This proves Proposition 2.PROOF OF PROPOSITION 5.4: The assertion of the proposition is true if

there exist constants M and r such that 0 < r < 1 and for T ∈ N

‖KT,j −KT+1,j‖ ≤ MrT+1, j = 0, 1, 2, . . . . (A.10)

Note now that KT,j (KT+1,j) is a solution to the matrix difference equation(5.9) at i = T − j (i = T + 1 − j) with the initial condition KT,T+1 = 0(KT+1,T+2 = 0). Subtracting (A.3) for KT,T−(i+1) from that for KT+1,T−(i+1),we have

KT,T−(i+1) −KT+1,T−(i+1) = B−1T−i(KT,T−i) −KT+1,T−i)AT−i

−B−1T−iKT,T−i)Q12,T−iKT,T−(i+1) +B−1

T−iKT+1,T−iQ12,T−iKT+1,T−(i+1).

Adding and subtracting B−1T−i ·KT,T−i ·Q12,T−i ·KT+1,T−(i+1) in the right hand

side gives

KT,T−(i+1) −KT+1,T−(i+1) = B−1T−i(KT,T−i) −KT+1,T−i)AT−i

−B−1T−i ·KT,T−i ·Q12,T−i(KT,T−(i+1) −KT+1,T−(i+1))

−B−1T−i(KT,T−i −KT+1,T−i)Q12,T−i ·KT+1,T−(i+1).

Rearranging terms yields

(I +B−1T−iKT,T−iQ12,T−i)(KT,T−(i+1) −KT+1,T−(i+1))

= B−1T−i(KT,T−i −KT+1,T−i)AT−i

−B−1T−i(KT,T−i −KT+1,T−i)Q12,T−iKT+1,T−(i+1).

From Proposition 5.3 it follows that the matrix

ZT,T−i = (I +B−1T−iKT,T−iQ12,T−i)

22

is invertible, then pre-multiplying the last equation by this matrix yields

KT,T−(i+1) −KT+1,T−(i+1) = Z−1T,T−i(B

−1T−i(KT,T−i −KT+1,T−i)AT−i

−B−1T−i(KT,T−i) −KT+1,T−i)Q12,T−iKT+1,T−(i+1)).

Taking the norms, using the norm property and the triangle inequality, we get

‖KT,T−(i+1) −KT+1,T−(i+1)‖

≤ ‖Z−1T,T−i‖ · (‖B

−1T−i‖ · ‖KT,T−i −KT+1,T−i‖ · ‖AT−i‖

+ ‖B−1T−i‖ · ‖KT,T−i) −KT+1,T−i‖ · ‖Q12,T−i‖ · ‖KT+1,T−(i+1)‖).

(A.11)

From (5.3) and (A.9) we have

‖KT,T−(i+1) −KT+1,T−(i+1)‖

≤

(

ab+1− ba

2

)

‖Z−1T,T−i‖ · ‖KT,T−i −KT+1,T−i‖

=1 + ba

2‖Z−1

T,T−i‖ · ‖KT,T−i −KT+1,T−i‖.

(A.12)

From the norm property and Golub, and Van Loan (1996, Lemma 2.3.3) weget the estimate

‖Z−1T,T−i‖ = ‖(I +B−1

T−iKT,T−iQ12,T−i)−1‖ ≤

1

1− ‖B−1T−iKT,T−iQ12,T−i‖

≤1

1− ‖B−1T−i‖ · ‖KT,T−i‖ · ‖Q12,T−i‖

By (A.9), we have

‖Z−1T,T−i‖ =<

1

1− 1−ba2

=2

1 + ba

Substituting the last inequality into (A.12) gives

‖KT,T−(i+1) −KT+1,T−(i+1)‖ < ‖KT,T−i −KT+1,T−i‖. (A.13)

Using (A.16) successively for i = −1, 0, 1, . . . , T − 1, and taking into accountKT,T+1 = 0 and KT+1,T+1 = B−1

T+2Q21,T+2 results in

‖KT,j −KT+1,j‖ < ‖KT,T+1 −KT+1,T+1‖ = ‖B−1T+2Q21,T+2‖

≤ ‖B−1T+2‖ · ‖Q21,T+2‖ ≤ b‖Q21,T+2‖, j = 0, 1, 2, . . . .

(A.14)

Recall that Q21,T depends on the solution to the deterministic problem (3.11),

i.e. Q21,T = Q21

(

x(0)T+1, x

(0)T , z

(0)T+1, z

(0)T

)

. From Hartmann (1982, Corollary 5.1)

and differentiability of Q21 with respect to the state variables it follows that

‖Q21,T ‖ ≤ C(α+ θ)T , (A.15)

23

where α is the largest eigenvalue modulus of the matrix A from (4.6), C is someconstant and θ is arbitrary small positive number. In fact, α+ θ determines thespeed of convergence for the deterministic solution to the steady state. Inserting(A.15) into (A.16), we can conclude

‖KT,j −KT+1,j‖ < bC(α+ θ)T+2, j = 0, 1, 2, . . . (A.16)

Denoting M = bC(α + θ) and r = α + θ we finally obtain (A.10). This provesthe proposition.

B Series expansion for Burnside’s model

Substituting (6.5) and (6.6) into (6.2) yields

y(0)(

x(0)t + σx

(1)t

)

+ σy(1)(

x(0)t + σx

(1)t

)

+ σ2y(2)(

x(0)t + σx

(1)t

)

+ · · ·

= βEt

exp[

θ(

x(0)t+1 + σx

(1)t+1

)]

[

1 + y(0)(

x(0)t+1 + σx

(1)t+1

)

+ σy(1)(

x(0)t + σx

(1)t

)

+ σ2y(2)(

x(0)t+1 + σx

(1)t+1

)

+ · · ·]

Expanding yt for small σ up to order two gives

y(0)t + σy

(0)1,t x

(1)t +

1

2σ2y

(0)2,t

(

x(1)t

)2

+ σy(1)x(0)t + σ2y

(1)1,t x

(1)t + σ2y

(2)t + · · ·

= βEt exp(θx(0)t+1)

[

1 + σθx(1)t+1 +

1

2

(

σθx(1)t+1

)2

+ · · ·

]

[

1 + y(0)t+1 + σy

(0)1,t+1x

(1)t+1

+ σ2 1

2y(0)2,t+1

(

x(1)t+1

)2

+ σy(1)t+1 + σ2y

(1)1,t+1x

(1)t+1 + σ2y

(2)t+1 + · · ·

]

Collecting the terms of like powers of σ of the last equation, we have

y(0)t + σ

[

(y(0)1,t x

(1)t + y(1)x

(0)t

]

+ σ2

[

y(2) + y(1)2,t x

(1)t +

1

2y(0)2,t

(

x(1)t

)2]

+ · · ·

= β exp(

θx(0)t+1

)

Et

(1 + y(0)t+1) + σ

[

θx(1)t+1(1 + y

(0)t+1) + y

(0)1,t+1x

(1)t+1 + y

(1)t+1

]

+ σ2[1

2(θx

(1)t+1)

2(1 + y(0)t+1) + y

(2)t+1 + y

(1)t+1x

(1)t+1 + y

(1)1,t+1x

(1)t+1 +

1

2y(0)2,t+1

(

x(1)t+1

)2

+ θy(0)1,t+1

(

x(1)t+1

)2

+ θx(1)t+1y

(1)t+1

]

+ · · ·

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